Although widely used in many countries, efficacy of orbital radiotherapy for treatment of Graves' ophthalmopathy has not been established in a prospective randomized double-blind controlled trial. It is also unclear if radiotherapy is more effective when given early in the course of the disease. Although currently believed to be rare, incidence of radiation retinopathy following radiotherapy for Graves' ophthalmopathy has not been defined with sensitive detection methods. Our specific aims are to determine: 1. If 20 Gy of external beam radiation directed to one orbit of a patient with Graves' ophthalmopathy results in improvement in one or more of the cardinal' expressions of Graves' ophthalmopathy in comparison with the untreated orbit when evaluated at three and six months after therapy. 2. If 20 Gy of external beam radiation delivered to the second orbit six months later in the course of the disease produces effects of equal magnitude to those observed when the first orbit was treated. 3. To relate the magnitude of the treatment effect to the time since onset of eye symptoms. 4. To determine whether any patients have findings characteristic of radiation retinopathy three years after orbital radiation therapy with 20 Gy. Study Design: Patients with mild to moderate Graves' ophthalmopathy who will be maintained euthyroid during the study will receive 20 Gy of supervoltage x-ray to one orbit. Six months later the other orbit will be treated. On entrance to the study and at three-month intervals for one year, inquiry will be made regarding patients' eye symptoms and function. Measurements will be made of proptosis, visual acuity, volume of extraocular muscle and retrobulbar connective tissue, extraocular muscle function, visual fields, relevant thyroid hormones and thyroid antibodies. Each evaluation will include, an orbital CT scan. Fluorescein angiography will be performed on admission and three years later. Between visits patients will be contacted regularly by telephone. Our long-term aim is to develop a rational, scientifically well supported approach to the treatment of Graves' eye disease.